# Full text of "Treatise On Analysis Vol-Ii"

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```434        XV    NORMED ALGEBRAS AND SPECTRAL THEORY

0. Now, by the definition of the adjoint (15.12.3), we see first that z e dom(#?)
and next that (x | H*  z + iz) = 0 for all x e doml/^), and since domC^) is
dense in E, we have H^ - z + iz = 0. Since dom(H) a dom^), we have
dom(#*) <=: dom(#*), and H* is the restriction of #*. Hence H*  z + iz = 0,
in other words z e E# . But E# c dom^) by definition, so that

0 = (z | (#* + iJ) - z) = ((# ! - il)  z | z) = ((H* - i/)  z | z) = -~2i||z||2,

and therefore finally z = 0.

To prove that F = E for all X such that */A > 0, we observe that if this is
true for 1 = A0, then the operator (H^  ^o/)""1 is everywhere defined and
continuous, and satisfies

by virtue of (15.13.10.1). It follows (15.12.11) that (Ht - A/)"1 is everywhere
defined and continuous for |A  /10| < |c/A0|. By induction, (H^  A/)"1 is
everywhere defined and continuous for |/l  (f)n/| < (f)w, and since the
integer n is arbitrary, we have shown that (Hl  /I/)""1 is everywhere defined
for all A in the half-plane JX > 0.

We have thus defined, for all A e C such that Ğ/A > 0, a continuous operator
A(X) = (^ - IT)"1 on Eğ with image equal to dom^) c dom(/P). Since
(H^  /l/)^(l) = /, it follows that, for any two complex numbers a, ft such
that */a > 0 and ,//? > 0, the operator

*(a, 0) = (^ - a/^O?) Ğ / + 08 - aX/0

is everywhere defined and continuous; also the operator A(p)(Hi  /?/) is the
identity on dom^), and therefore

(15.13.10.2)          K(oc9 0)K(p, y) = K(a, 7),       K(a, a) == /.

Hence the operators K(a, fi) are invertible and bicontinuous.

This being so, for every x e dom(H*) we have K(i, X)  x e dom(H *) and

(H* - II) - (K(i9 X)-x) = (H* - A/) - jc + (A - i

and likewise (J^T* - //)  (JC(A, 0  x) = (/?* - A/)  x. This shows that K(i, A)
is a bicontinuous bijection of EjJ onto E+(A). Similarly for E~(A). we
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